First, wave functions are complex, that is they involve the square root of minus one. They determine the probability, not just for position but for momentum and other "observables" by a simple mathematical operation that turns the complex number into its "complex square", a real number that can be identified with a probability. Sorry if that seems over complicated, but it's important to be clear about wave functions if you're going to think about quantum mechanics.

Linear in this case means you can add them and subtract them. Sometimes it helps to think of the analogy of sound waves where you can add tones (say C,E, and G on the piano) to make a new sound, a chord. You can add wave functions to make a new wave function, called a superposition, and this is a basic tool of quantum mechanics. So quantum mechanics is called a linear theory, because it relies so basically on this linear property of the wave functions.

There are a lot of open questions in advanced quantum theory, and people try all kinds of modifications to see if they can answer those questions. One of the ways they have done this is to abandon the linearity, to try to use wave functions that don't add, and therefore don't form superpositions, at least not in the old simple way. So far this research program is still alive, but without superposition its hard to find things out, so it doesn't have a lot to show. Linear QM is a tremendoiusly successful theory, that explains all kinds of weird things the experimenters have come up with.